Lattice-type reactive quadripoles



Dec. 7, 1954 R. P. LEROY 2,696,591

LATTICE-TYPE REACTIVE QUADRIPOLES Filed Dec. 20. 1951 5 Sheets-Sheet 1mvsw-roR ROBERT PIERRE LEROY Dec. 7, 1954 R. P. LEROY 2,696,591

LATTICE-TYPE REACTIVE QUADRIPOLES Filed Dec. 20, 1951 5 Sheets-Sheet 2(m+m=m+n.3)

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ENTOR:

RoBERT PIERRE LERoy wy i Dec. 7, 1954 R. P. LEROY 2,696,591

LATTICE-TYPE REACTIVE QUADRIPOLES Filed Dec. 20. 1951 5 Sheets-Sheet :5

RoBERT PIERRE LEROY 1954 R. P. LEROY 2,696,591

LATTICE-TYPE REACTIVE QUADRIPOLES Filed Dec. 20, 1951 5 Sheets-Sheet 4IN E/v To A ERT P/ERR LEROY Dec. 7, 1954 R. P. LEROY 2,696,591

LATTICE-TYPE REACTIVE QUADRIPOLES Filed Dec. 20. 1951 5 Sheefcs-Sheet 5ROBERT PIERRE LEROY United States Patent LATTICE-TYPE REACTIVEQUADRIPOLES Robert Pierre Leroy, Paris, France, assignor to Compagnieindustrielle dos Telephones, Paris, France, a French corporationApplication December 20, 1951, Serial No. 262,599

Claims priority, application France December 28, 1950 4 Claims. (Cl.333-74) In French Patent No. 949,926, filed on August 23, 1945,applicant showed how, by using lattice type networks with three separatereactances, it is possible to obtain filters equivalent to the classicfilters with a single pass band (band-pass, low-pass, high-pass).

The object of the present invention is the application of the samemethod to more general types of filters, independent of the research onfilters comprising several pass-bands, this class of filter has to beconsidered if it be desired to eliminate from a standard type offiltersection identical impedances which are in series or in shunt ineach of the branches of the filter-section, by employing a methodemployed by Mason, for example in his work Electromechanical Transducersand Wave Filters.

The transformed filter-section is not usually a singlepass-band filtersection when such is the case of the original section.

The separate transformation of the filter-section therefore requires theextension of the method employed in the above-mentioned patent.

In the following it is assumed that m and zz have neither zeros norpoles in common-except in certain cases for zero or infinitefrequenciesthat is to say that the impedance function has no criticalpoints outside the branching points. This restriction is not essential,as has been shown in the above-mentioned French patent (page 15, lines11 et seq.).

Let us first of all assume a conventional lattice type network whichdoes not pass either the Zero frequency or the infinite frequency.

It may be assumed that the branches Z1 and 22 allow a zero at theorigin, because, if this were not so, we should be brought back to thiscase by taking the filter-section of inverse branches,

At infinity, Z1 and Z2 simultaneously offer a zero or a pole. We willassume that they do not offer them simultaneously anywhere else.

- Therefore these two cases have to be considered:

The method of determination of the filter-network according to theinvention is based, as in the above-mentioned patent, on the research onsolutions by reactances x1, m of the equation:

if it be desired to substitute the branches x1, x2 for the equalbranches Z11 or of the equation:

id- 1 Z+ 1 1-1- 2 if it be desired to substitute for the equal branchesz2 the branches S1, S2.

2,696,591 Patented Dec. 7, 1954 0m n being a polynomial in p of whichthe coefiicient of highest power is equal to 1 and of degree mn atmost.This form is necessary in order that x1+z2 and x2+zz admit for the polesof Z2 the same remainders as Z2 or z1+z2. The condition mn 0 istherefore necessary; the branch of lowest degree cannot be replaced.

We have m-n-l-l parameters available to reduce the degree of xr-l-zz orxz-l-zz, by causing common factors on the numerator and on thedenominator to disappear by a suitable choice of k 0mn; in the mostfavourable case, we can expect a reduction of (mn+1) resonant circuits.However, it is obviously necessary for x1 and x: to retain theircharacter of reactances.

The boundaries of the 5 pass bands are the frequencies Their number is:2m+2n2 r2v2=2[3 and the reductions of degree possible on x1 and on am bya suitable choice of [C203 causing the appearance of one or more commonfactors on the numerator and denominator of x1 and x2, are obtainedwhen:

As we have (s+1) parameters, we will try to express that the equation (p=0, of degree fi+s in p admits s+1 of the m+nl roots of Wm+n1(p =0.

If k fis has been thus determined, the degrees of x1 and x2 will intotal be reduced by (s-l-l) and if x1 and x2 again represent reactances,we obtain a gain of (s+l) of the resonant circuits.

Whatever the polynomial (Mp (p admits for k =Othe zeros of Pm1uQm-y and,when k increases" 3 the roots of (p are displaced inside the pass-bands,where the following condition is fulfilled:

The roots of (p are therefore shifted towards the zeros of Wm+1L1(pwhich are the nearest of the zeros of Pm--ltQmy, and in the pass-bandsof which these zeros define a boundary. If each band contains at leasttwo zeros of Wm+1z+1(p it is certain that each of the 18+s zeros of (pis shifted towards a diiferent root; if 93 be chosen in such a Way that,by causing k to increase, (p admits for a value [6 m simultaneously s+1of the B+s zeros of- Wm+n 1(p the values of x1+z2 and xz-l-zz given bythe Formulae 6 will always represent reactances, if none of the roots ofthe numerators is passed for k km by zero or infinite values, that is tosay, if km be less than and than ko =1.

The same applies, under these conditions to x1 and x2, which show thepecularities of z2 With the same remainders.

In the case under consideration, a. choice has to be made from the B-l-spossible roots; thus we should have a priori tests; this number can bereduced in the case of pass bands of which the boundaries arerespectively zeros of Put-1+ and (2712-, and only contain a zero of Wm+n1; in this case a single zero of Wm+n-l may be eliminated by two rootsof (p the number of such bands is obviously at most equal to thesmallest of the numbers m1-,u, m-u;- the reduction of the zeros ofWm+1L--1 to be considered is therefore m1 at most; there remaintherefore at least- Generally speaking we shall designate by s+s' thenumber of zeros of Wm+n1 which can be used a priori: l s'f3.

We will designate by one of the sets of zeros of Wm+n+1 with which weare seeking to merge s+1 of the zeros of ('p we will put P m-l-[JQm-P)nvl ny P =-rflx 10,,20, will be determined by the Lagrange formula:

In order that this formal solution may apply, it is first of allnecessary that for the other zeros of with which the zeros of (p mightbe merged we should have:

condition of alternation.

In other words, 1f we conslder the curve which passes through the pointsfor h=-1', 2, s+1 aswell as its symmetricalin relation to the axis ofthe frequencies, these curves should not contain inside themany oftheother points Qm) n 0 we shall obtain by the Formulae 6 the limbs of.afilter network showing a gain o'f'(s+1); circuits over the conventional filter-section, which gain is obtainedon the. one branch orthe other according to the sign of the corresponding 5.

It will be seen that generally speaking, 21 can be obtained by puttingin shunt a capacity, an inductance and m-l resonant circuits; 12 can beobtained by putting. in.

shunt a capacity, an inductance and (Yb-1 resonant circuits (Fig. 1).

x1 and x2- may be represented by a diagram of the same typewith m and mresonant circuits (-Fig. l) insuch a way that determining x1 and x2 bythe Formulae 6- and 10 under the Conditions l2, l3, 14.

A gain of (s+l) resonant circuits orof (s-|--l-)- quartz can thus beobtained with respect to the conventional filter-section if the aboveconditions are' fulfilled.

A simple case is that in which s=m-n=1 the polynomials [c 68 thenrepresent the straight lines as a func-- tion of p and it is certainthat by taking the E of the same sign-positive since 05 is defined tothe nearest signit is possible to choose-in one. way at least-the twozeros of Wm+n so as to fulfill the Conditions 12.

Then we only have to verify 13 and 14 which become, the zeros beingdesignated by -a12 and -r2 Z More precisely, if we consider the pointsA1, A2 As+1 of co-ordinates we shall select from themthose whichdetermine a convex broken line leaving all the other points. on the sideopposite to the axis of the 12 and we shall retain among: the segmentsof this line those of which the slope-is lower. in absolute value than 1(13') and theordinate at the origin less than (14) in absolute value.

There are physically acceptable solutions corresponding' to them-Whichpermit a: gain of two resonant circuits.

in which has the same significance as in the Case 1 and consequently:

hThe factors of reduction for an and x2 are obtained w en:

admits roots of Wm+n(p )=0.

The 8+s zeros of 5, given by Pm- Qm-v for k =0, penetrate into the passbands when It increases as it approaches the Zeros nearest to Wm+n(p Thenumber of zeros of Wm+n is comprised between fl+s and the largest of thenumbers m,u, m-u, in the circumstances similar to those encountered inCase 1.

If ,.=v=n, it would seem that the minimum is reduced to s. In this case:

The pass bands are mn in number and must contain m+n zeros; one at leastof these bands therefore contains several of them, so that the lowerlimit is s+l and not s.

As in the previous case, we have to choose between s+s' roots with 1 53.

Putting:

G(p being defined by:

The an must be chosen in such a way that These conditions ensure thatthe expressions appearing in the left hand sides of (17) representreactances; the same applies to their inverse functions; in order thatit may be the same for x1 and x2 it is necessary and it is sufficientthat their remainders to infinity be positive, that is to say that, inaddition to (25) we have:

Under these conditions, z1 and Z2 being obtainable by putting aninductance in series with the grouping in parallel of a capacity, aninductance, and of m1 or nl resonant circuits (Fig. 2) the reactances x1and x2, which may be substituted for the two branches z1 allow the sametype of structure with m and m" resonant circuits, m+m being equal to2(m1)-(s+l)=m+n3 (Fig. 2').

When s: l, remarks similar to those of Casel can be ma c.

It remains to consider the filter-sections which allow one or the otherto pass, or one or the other of the zero and infinite frequencies.

In the second hypothesis, it is suflicient to deal with the case of theinfinite frequency, as the case of the zero frequency can be inferredtherefrom by substituting 1 for P p I. Filter-sections allowing the zeroand infinite frequencies to pass It can be assumed that the branch Z1.has zeros at the origin according to their behaviour at infinity we havethe Cases 10: and 1/3:

I a P,

x1+z2, x2+z2 may also be acceptable, for 12:0, p=, hence:

x1+z2, xz-I-zz to be reactances in order that the same may apply for x1and m. We are again lead to write:

The reduction of degree will be obtained by determining km 0s-1 so as tocause s zeros of to coincide with s zeros of Wm+n+2, selected from s+s'(s' provided that, 08-1 being fixed, no coincidence occurs for a valueof k lower than km (conditions of alternation).

7 the conditions of alternation: Z-i a1( 'z +'h) I 5 2311 beingsufficient, since, according to their form, the numerators of the righthand sides of (29) cannot take zero or infinite roots.

Here (Fig. 3), z1 is composed of the putting in parallel of aninductance, a capacity and m resonant circuits; z2 being formed by theplacing in series with an inductance of the grouping in parallel of nresonant circuits and a capacity.

The branches x1 and are substituted for the branches Z1 are similar instructure to that of n with respectively. m and m" resonant circuits, mm" being equal to 2ms=m+n (Fig. 3).

REPLACEMENT OF 2:

and proceed as before.

In addition to conditions of alternation it is necessary here that:

s1 and s2 then have the same type of structure as Z2 with n and n'resonant circuits, n i-n being equal to m+n (Fig. 3").

In brief, if Im-nIZl, gain of Im-nl resonant circuits by replacement ofthe branches of the highest degree.

CASE 1 m can be obtained by placing an inductance in series with thegrouping in parallel of a capacity, an inductance, and of (m'l) resonantcircuits; 12 can be obtained by placing in parallel n resonant circuitsand a capacity (Fig. 4).

REPLACEMENT OF e,

It will be seen that, by proceeding as before, if We fulfil theconditions of alternation as well as km 1, we obtain, on x1 and x2, atotal gain of s resonant circuits in relation to the branches 2.1. Thenumber of resonant circuits in x1 and x2 is in all:

If we can satisfy, in addition to the conditions of al ternation:

we can obtain on s1 and s2 a gain of (n-m) resonant circuits in relationto the branches 22. They have the same type of structure as zz withm-l-n resonant circuits in all (Fig. 4").

II. Filter-sections allowing the infinite frequency to pass and not thezero frequency It can be assumed that 21 and Z2 allows zeros at theorigin: the one will admit a pole at infinity, the other:

a zero. We can then put:

1- IP m 11 can be obtained by putting an inductance in series with thegrouping in parallel of a capacity, an inductance and of m-l resonantcircuits, zz can be obtained by placing in parallel an inductance, acapacity and (n-1) resonant circuits (Fig. 5).

REPLACEMENT OF 2 We find:

1 s [Qm+ns. .1 1+ 2 P m+n(P 1 n[Qm n s] x1 and x2 given by:

x1 and x2 have the type of structure of zi, but with m and m" resonantcircuits (Fig. 5')

being determined by the same method as before.

Necessary and sufficient conditions: of alternation,

m 1, mm. 104%) REPLACEMENT 0F 22 In this case we have:

s1 and s2 given by:

n in man-'1 o What is claimed is:

1. A lattice-type, electrically symmetrical reactive fourtermi-nalnetwork comprising a number 6 of pass bands excluding zeroand infinitefrequencies, said network being equivalent to a band-pass filter withtwo sets of opposite and identical branches consisting, respectively, ofreactances 21 and zz given by the expressions wherein A1 and A2 areconstants, and p designates the the quantity 2117'), 1 representing theoperating frequency, R, S, U and V are polynomials in terms of 12 and m,m, n, n, u and v are integral parameters, the differences m-m' and n-nbeing equal to each other and capable of assuming either of the valuesand 1 but no other, n having a value between 1 and m inclusive, saidnetwork being characterized by the fact that it comprises a set of twoopposite branches equal to zz and a set of two opposite, unequalbranches x1 and x2 given by the expressions being a polynomial of theorder s in terms of p whose highest-order term has the coefiicient andwhich is defined by the equation 8+1 G 2 m (2 )=2 h li' h=1 2 k2)- dp pa'h wherein sh=i1 and wherein o'1 ,a2 -0's+1 represent s+1 zeros ofzr-I-zz selected from a maximum number of s-f-B thereof, and whereinfurther G(p is equal to the product H (P+ i) h=1 and the values of aregiven by "g l M'IIQ7II I wherein A1 and A2 are constants, p designatesthe quantity 21rjf, 1 representing the operating frequency, P, Q, R, S,U, and V are polynomials in terms of p and m, n, ,u. and v are integralparameters, said network comprising a first pair of identical andopposite branches as well as a second pair of unequal and oppositebranches having the following characteristics: (a) if m-n=s 1, the

branches of said first pair are equal to zz and the branches x1, am ofsaid second pair are given by the expressions being a polynomial of theorder s-l in terms of p whose highest-order term has the coefficientrepresent s zeros of z1+z2 selected from a maximum of s+,8 thereof, andwherein further G(p is equal to the product 8 II (p+ i) h=1 and thevalues of are given by the branches x1 and x2 being realizable by meansof a number of resonant circuits falling short by s of the number ofsuch circuits in the identical and opposite branches n of saidequivalent filter; and (b) if nm=s l, the branches of said first pairare equal to z1 and the branches yi, y: of said second pair are given bythe expressions being a polynomial as defined by the foregoing equationwherein, however, the values of are given by wherein A1 and A2 areconstants, p designates the quantity 21rjf, 7 representing the operatingfrequency, P, Q, R, S, U and V are polynomials in terms of 12 and m, n,n and 11 are integral parameters, said network comprising a first pairof identical and opposite branches as well as a second pair of unequaland opposite branches number of s+fi-l thereof, and wherein further G(pis equal to the product the branches x1 and xz being realizable by meansof a number of resonant circuits falling short by s of the number ofsuch circuits in the identical and opposite branches Z1 of saidequivalent filter; and (b) if nm= 1, thebranches of said first pair areequal to 1 and the branches y1, yz of said second pair are given by theexpression wherein, however, the values of are given by A2 lib] A11muQmv P ='h the branches yr and yz being realizable by means of a numberof resonant circuits falling short by s of the number of such circuitsin the identical and opposite branches zz of said equivalent filter.

4. A lattice-type, electrically symmetrical reactive four-terminalnetwork comprising a number of pass bands including the infinitefrequency but not the zero frequency, said network, being equivalent toa bandpass filter with two sets of opposite and identical branchesconsisting, respectively, of reactances Z1 and Z2 given by theexpressions and wherein A1 and A2 are constants, p designates thequantity 21rj 1 representing the operating frequency, P, Q, R, S, U andV are polynomials in terms of p and m, n. p, and 1 are integralparameters, said network comprising a first pair of identical andopposite branches as well as a second pair of unequal and oppositebranches having the following characteristics: (a) if s=mn 0,

the branches of said first pair are equal to z2- and the branches x1, x2of said second pair are given by the expressions ewktsn rtp being apolynomial of the order s in terms of p whose highest-order term has thecoeflicient and which is defined by the equation 8+1 G 2 WAP FEM? p +awherein eh=ztl and wherein 0'1 e2 as represents s zeros of 1 and zzselected from a maximum number of s+B thereof, and wherein further G(pis equal to the product and the values of 2: h are given by the branchesx1 and am being realizable by means of a a number of resonant circuitsfalling short by s-l-l of the number of such circuits in the identicalandopposite branches Z1 of said equivalent filter; and (b) if s=n-m 1,the branches of said first pair are equal to z1 and the branches yr, yzof said second pair are given by the expressions being a polynomial ofthe order s-l in terms of 11 whose highest-order term has thecoefiicient and which is defined by the equation 8 G 2 l a-1(P )=2 hfl W(p -Wimp P P=wwherein eh=i1 and wherein 0'1 o'2 o'S represent s zeros ofZ1 and 22 selected from a maximum number of s+/31 thereof, and whereinfurther G(p is equal to the product 7 the branches yr and ya beingrealizable by means of a number of resonant circuits falling short by sof the number of such circuits in the identical and opposite branches zzof said equivalent filter.

References Cited in the file of this patent FOREIGN PATENTS NumberCountry Date 949,926 France Mar. 14, 1949

